Titlebook: Lectures on Mathematical Theory of Extremum Problems; Igor Vladimirovich Girsanov,B. T. Poljak Book 1972 Springer-Verlag Berlin · Heidelbe

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Sufficient Extremum Conditions. Examples,We now apply the results of the preceding lecture to various problems.

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Introduction,systematized and brought together under the heading of the ., with its innumerable applications to physics and mechanics. Attention was devoted principally to the analysis of . and . defined over the entire space or restricted to some smooth manifold. The extremum conditions in this case are the . (

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Supporting Hyperplanes and Extremal Points,e closed hyperplane . is called a . for A at the point x.. The geometric sense of a supporting hyperplane is quite simple: the set A lies on one side of the hyperplane and cuts it in one point x. (Fig. 5).

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Calculation of Dual Cones,plication of the Dubovitskii-Milyutin theorem to determine necessary conditions for an extremum, it remains to show how one constructs dual cones, This we now proceed to do. Some results in this connection were presented in Lecture 5 (Lemmas 5.2 to 5.10).

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Sufficient Extremum Conditions,re also sufficient. Of course, elementary examples show that in general this is not true. Nevertheless, we shall prove that, under certain additional assumptions, the necessary extremum conditions are also sufficient, in an important class of extremal problems — convex problems. Sufficient condition